Articles

AbstractThis paper presents and compares five analytical formulas for the approximation of stop-loss premiums. Two of them, based on the inverse Gaussian distribution, are not widely known. The authors also suggest a technique which improves the precision of these approximations for portfolios.

AbstractA simple extension of the method of Kornya is derived. The extended method applies to the convolution of triatomic distributions with nonnegative support while the original method is restricted to diatomic distributions. This way, the algorithm can be applied in the calculation of the distribution of the total claims of a pension fund where only death and disability of active members are considered.

AbstractThe efficiency of the (new) Swiss bonus-malus system is analyzed according to three efficiency measures. In addition, the analytical form of the stationary distribution of the system, which is involved in two of the efficiency measures, is obtained as a byproduct of its recursive calculation scheme.

AbstractWe consider a family of aggregate claims processes that contains the gamma process, the Inverse Gaussian process, and the compound Poissonprocess with gamma or degenerate claim amount distribution as special cases. This is a one-parameter family of stochastic processes. It is shown how the probability of ruin can be calculated for this family. Extensive numerical results are given and the role of the parameter is discussed.

AbstractThe classical model of collective risk theory is extended in that a diffusion process is added to the compound Poisson process. It is shown that the probabilities of ruin (by oscillation or by a claim) satisfy certain defective renewal equations. The convolution formula for the probability of ruin is derived and interpreted in terms of the record highs of the aggregate loss process. If the distribution of the individual claim amounts are combinations of exponentials, the probabilities of ruin can be calculated in a transparent fashion. Finally, the role of the adjustment coefficient (for example, for the asymptotic formulas) is explained.

AbstractIt is shown how user friendly examples of ruin theory problems can be constructed, i.e., examples where all the parameters are integers or rational numbers. Tables containing more than 200 such examples are provided.

AbstractThe aggregate claims process is modelled by a process with independent, stationary and nonnegative increments. Such a process is either compound Poisson or else a process with an infinite number of claims in each time interval, for example a gamma process. It is shown how classical risk theory, and in particular ruin theory, can be adapted to this model. A detailed analysis is given for the gamma process, for which tabulated values of the probability of ruin are provided.

AbstractThe first method, essentmlly due to GOOVAERTS and DE VYLDER, uses the connection between the probabiliy of ruin and the maximal aggregate loss random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distributions. Then the probability of ruin can be calculated in a transparent fashion; the main problem is to determine the nontrivial roots of the equation that defines the adjustment coefficient. For the third method one observes that the probability of ruin is related to the stationary distribution of a certain associated process. Thus it can be determined by a single simulation of the latter. For the second and third methods the assumption of only proper (positive) claims is not needed.

Dufresne F. & Gerber H.U. (1988). The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics, 7(3), 193-199. [url] [abstract]

AbstractIn the classical compound Poisson model of the collective risk theory we consider X, the surplus before the claim that causes ruin, and Y, the deficit at the time of ruin. We denote by f(u; x, y) their joint density (u initial surplus) which is a defective probability density (since X and Y are only defined, if ruin takes place). For an arbitrary claim amount distribution we find that f(0; x, y) = ap(x + y), where p(z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f(u; x, y) can be calculated explicitly, if the claim amount distribution is exponential or, more generally, a combination of exponential distributions. We are also interested in X + Y, the amount of the claim that causes ruin. Its density h(u; z) can be obtained from f(u; x, y). One finds, for example, that h(0; z) = azp(z).

AbstractIn the classical compound Poisson model of the collective theory of risk let ?(u, y) denote the probability that ruin occurs and that the negative surplus at the time of ruin is less than ? y. It is shown how this function, which also measures the severity of ruin, can be calculated if the claim amount distribution is a translation of a combination of exponential distributions. Furthermore, these results can be applied to a certain discrete time model.

Actes de conférence (partie)

Viquerat S. & Dufresne F. (2008). How to get rid of round-off errors in recursive formulas. Insurance: Mathematics and Economics.

AbstractAbstract¦This paper shows how to calculate recursively the moments of the accumulated and discounted value of cash flows when the instantaneous rates of return follow a conditional ARMA process with normally distributed innovations. We investigate various moment based approaches to approximate the distribution of the accumulated value of cash flows and we assess their performance through stochastic Monte-Carlo simulations. We discuss the potential use in insurance and especially in the context of Asset-Liability Management of pension funds.